Square roots of negative numbers

You went from:

sqrt(48)

to

4*sqrt(12)

When I pull something out from under a radical like that (take a perfect square out), I imagine there's like this force field wall thing that does a square-root function on it when it's on the way out. (Or, if I'm pushing something into the radical from outside, the force-field squares it when it goes through that way.).

When you have that 48, you might have thought, "That's divisible by 4, a perfect square!" and you'd have been right, but when you pull it through the force field, it square-roots it, so you'd have 2*sqrt(12), not the 4*sqrt(12) that you got.

(You can enter two expressions on a calculator to see what's equivalent to what.)

So... how do you get to 4*sqrt(3) ?

Well, I see that they're equivalent, because pushing that 4 back through the radical sign force-field and squaring it makes it 16, which times 3, makes the 48.

So I needed to see the perfect square 16 (which is 4 squared) inside 48, not 4, to fully simplify it. 2*sqrt(12) is also equivalent, but you can still pull another "2^2" (or 4) out of it.

You can practice here on some problems with positive numbers inside the radical, or review the content in that area. It's just more of the same with negative numbers if you get the concept of i and removing it, which you seem to.

(Don't worry about the force-field thing if it doesn't work for you. I've just found it helpful to visualize it in that way, once I got the knack of why it's that way.)